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Introduction to Linear Polynomials — Class 9 Mathematics

Ganita Manjari · Chapter 2

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Chapter summary

This chapter builds on algebraic expressions to define univariate polynomials — expressions in a single variable such as — and their degree, the highest power of the variable. Polynomials of degree are called constant, linear, quadratic and cubic respectively. The focus is the linear polynomial, of the form . A linear pattern is a sequence whose consecutive terms differ by a constant amount, and many real situations (taxi fares, savings, growing tile patterns) follow such patterns. When a linear polynomial is set equal to a constant, a linear equation results. Two variables related linearly satisfy : here is the slope and the -intercept. Plotting two points and joining them graphs the line. Changing tilts the line (steeper when ); changing shifts it up or down, keeping lines of equal slope parallel. Positive slope models linear growth, negative slope models linear decay. The chapter also treats polynomials as input-output (function) machines.

Polynomials and degreeLinear polynomialsLinear patternsGrowth and decayGraphing \(y=ax+b\)

Key ideas & terms

Polynomial: An algebraic expression in one variable made of terms with whole-number powers, such as .
Degree: The highest power of the variable in a polynomial; e.g. has degree .
Linear polynomial: A polynomial of degree one, of the form with .
Slope: The coefficient in ; it measures the steepness of the line.
\(y\)-intercept: The constant in ; the point where the line meets the -axis.
Linear pattern: A sequence in which the difference between consecutive terms is constant.

Extra questions & answers

What is the degree of the polynomial ?

Its degree is , the highest power of , so it is a quadratic polynomial.

Find the value of the linear polynomial when .

Substituting gives .

What do and represent in the line ?

is the slope of the line and is the -intercept where the line cuts the -axis.

The sum of two numbers is and one is more than the other; find them.

If the smaller is , then , so , ; the numbers are and .

Does linear growth have positive or negative slope?

Linear growth is shown by a line with positive slope; linear decay has negative slope.

Through which point does every line pass?

Every such line passes through the origin , since when .

Find and if gives at and at .

Subtracting the equations gives , so and : .

What is the nth term of the tile pattern ?

The th term is , a linear polynomial with common difference .

How are two lines with equal slope but different intercepts related?

They are parallel to each other, since only the -intercept differs while the slope is fixed.

A phone costing 10000 loses 800 per year; what models its value after years?

Its value is , a linear decay with negative slope .

Think and Reflect

1. Can you identify the terms, variables and coefficients of this algebraic expression?

The algebraic expression is . The terms are , and . The variables are and . The coefficient of is , the coefficient of is and the coefficient of is . There is no constant term.

2. How is it different from the algebraic expression in Example 1?

The expression in Example 1 is which has two terms containing and separately and a constant term , and no product term of the two variables. The expression has a product term containing both variables multiplied together, so its value depends on the product (the area), not only on and separately.

Think and Reflect

1. Can you identify the terms, variables and coefficients of this algebraic expression?

The expression is or . The terms are and . The variable is only. The coefficient of is and the coefficient of is . There is no constant term.

2. Can you point out any similarity or difference between the algebraic expressions obtained in Examples 1 and 3?

Similarity: Both are algebraic expressions involving variables, coefficients and operations. Difference: Example 1 involves two variables and and is of degree 1 in each variable. Example 3 involves only one variable and has degree 2 because of the term ; it is a quadratic polynomial while Example 1 is not a univariate polynomial.

Exercise Set 2.1

1. Find the degrees of the following polynomials: (i)

The highest power of is , so the degree is .

1. Find the degrees of the following polynomials: (ii)

The highest power of is , so the degree is .

1. Find the degrees of the following polynomials: (iii)

This can be written as , so the degree is . It is a constant polynomial.

1. Find the degrees of the following polynomials: (iv)

The highest power of is , so the degree is . It is a linear polynomial.

2. Write polynomials of degrees 1, 2 and 3.

Examples: Degree 1: . Degree 2: . Degree 3: . Any polynomial with the correct highest power is acceptable.

3. What are the coefficients of and in the polynomial ?

The coefficient of is and the coefficient of is .

4. What is the coefficient of in the polynomial ?

There is no term, so the coefficient of is .

5. What is the constant term of the polynomial ?

The constant term is .

Think and Reflect

Find the perimeter of squares with sides 1 cm, 1.5 cm, 2 cm, 2.5 cm and 3 cm. What will happen to the perimeters if the sides increase by 0.5 cm?

Perimeter . For cm, cm; cm, cm; cm, cm; cm, cm; cm, cm. Each increase of cm in side increases the perimeter by cm. This constant difference shows a linear pattern.

Think and Reflect

If a player paid , how many matches did he play?

Total cost is . So and . He played 11 matches.

Think and Reflect

We have learnt that to evaluate the value of an algebraic expression, we substitute a value of the variable in the given expression. Consider Example 3, where the wire is bent to form a rectangle. Here, the area of the rectangle, , is a function of . Can you interpret this as an input-output process? What value does the expression take when cm?

Yes, is a function where input is the length and output is the area. For cm, the width is cm and . So the output is .

Exercise Set 2.2

1. Find the value of the linear polynomial if: (i)

1. Find the value of the linear polynomial if: (ii)

1. Find the value of the linear polynomial if: (iii)

2. Find the value of the quadratic polynomial if: (i)

2. Find the value of the quadratic polynomial if: (ii)

2. Find the value of the quadratic polynomial if: (iii)

3. The present age of Salil's mother is three times Salil's present age. After 5 years, their ages will add up to 70 years. Find their present ages.

Let Salil's age be , mother's age . After 5 years: . So , , . Salil is 15 years and his mother is 45 years.

4. The difference between two positive integers is 63. The ratio of the two integers is 2:5. Find the two integers.

Let numbers be and . Then , , . Numbers are and .

5. Ruby has 3 times as many two-rupee coins as she has five rupee-coins. If she has a total , how many coins does she have of each type?

Let five-rupee coins be , two-rupee coins . Value: . So , . She has 8 five-rupee coins and 24 two-rupee coins.

6. A farmer cuts a 300 feet fence into two pieces of different sizes. The longer piece is four times as long as the shorter piece. How long are the two pieces?

Let shorter be , longer . , , . Shorter is 60 feet, longer is 240 feet.

7. If the length of a rectangle is three more than twice its width and its perimeter is 24 cm, what are the dimensions of the rectangle?

Let width , length . Perimeter . So , , cm, cm.

Think and Reflect

Predict the number of squares in the next three stages of the pattern and write the sequence of numbers up to Stage 7 of the pattern.

Stage 1 has 1, Stage 2 has 3, Stage 3 has 5, Stage 4 has 7. Adding 2 each stage, Stage 5 has 9, Stage 6 has 11, Stage 7 has 13. Sequence is .

Think and Reflect

Using the expression , can you find out how many tiles will be there in the 15th stage and the 26th stage of the pattern? Also, which stage will contain 21 tiles and 47 tiles?

For , tiles. For , tiles. Solve gives . Solve gives .

Think and Reflect

What amount will be left on the 15th day? How many days will it take for the entire amount to be spent?

Amount left after days is . On 15th day: rupees. Entire amount spent when , so days.

Think and Reflect

For how many km will the fare be ?

Fare for km is . Set , , km.

Exercise Set 2.3

1. A student has in her savings bank account. She gets every month as pocket money. How much money will she have at the end of every month from the second month onwards? Find a linear expression to represent the amount she will have in the month.

After 1st month: , 2nd month: , 3rd month: etc. General expression: rupees in the month.

2. A rally starts with 120 members. Each hour, 9 members drop out of the group. How many members will remain after 1, 2, 3, ... hours? Find a linear expression to represent the number of members at the end of the hour.

After 1 hour: , 2 hours: , 3 hours: . Expression: members after hours.

3. Suppose the length of a rectangle is 13 cm. Find the area if the breadth is (i) 12 cm. Find the linear pattern representing the area of the rectangle.

Area = . General pattern: where is breadth, linear in with slope 13.

3. Suppose the length of a rectangle is 13 cm. Find the area if the breadth is (ii) 10 cm. Find the linear pattern representing the area of the rectangle.

Area = . Pattern .

3. Suppose the length of a rectangle is 13 cm. Find the area if the breadth is (iii) 8 cm. Find the linear pattern representing the area of the rectangle.

Area = . Pattern , increases by 13 for each 1 cm increase in breadth.

4. Suppose the length of a rectangular box is 7 cm and breadth is 11 cm. Find the volume if the height is (i) 5 cm. Find the linear pattern representing the volume of the rectangular box.

Volume = . Pattern where is height, linear with slope 77.

4. Suppose the length of a rectangular box is 7 cm and breadth is 11 cm. Find the volume if the height is (ii) 9 cm. Find the linear pattern representing the volume of the rectangular box.

Volume = . Pattern .

4. Suppose the length of a rectangular box is 7 cm and breadth is 11 cm. Find the volume if the height is (iii) 13 cm. Find the linear pattern representing the volume of the rectangular box.

Volume = . Pattern , increases by 77 for each 1 cm increase in height.

5. Sarita is reading a book of 500 pages. She reads 20 pages every day. How many pages will be left after 15 days? Express this as a linear pattern.

Pages left after days: . After 15 days: pages left.

Think and Reflect

What is the cost for travelling 15 km? For how many kilometres will the cost of the journey be ?

Cost function . For , rupees. Solve , , km.

Think and Reflect

What will be the height of the water at the end of 5 months?

Height . For , m.

Exercise Set 2.4

1. Suppose a plant has height 1.75 feet and it grows by 0.5 feet each month. (i) Find the height after 7 months.

. After 7 months: feet.

1. Suppose a plant has height 1.75 feet and it grows by 0.5 feet each month. (ii) Make a table of values for varying from 0 to 10 months and show how the height, , increases every month.

:0→1.75, 1→2.25, 2→2.75, 3→3.25, 4→3.75, 5→4.25, 6→4.75, 7→5.25, 8→5.75, 9→6.25, 10→6.75 feet. Each month adds 0.5 feet.

1. Suppose a plant has height 1.75 feet and it grows by 0.5 feet each month. (iii) Find an expression that relates and , and explain why it represents linear growth.

. It is of form with . Equal increase each month shows linear growth with positive slope 0.5.

2. A mobile phone is bought for . Its value decreases by every year. (i) Find the value of the phone after 3 years.

. After 3 years: rupees.

2. A mobile phone is bought for . Its value decreases by every year. (ii) Make a table of values for varying from 0 to 8 years and show how the value of the phone, , depreciates with time.

:0→10000,1→9200,2→8400,3→7600,4→6800,5→6000,6→5200,7→4400,8→3600 rupees. Decreases by 800 each year.

2. A mobile phone is bought for . Its value decreases by every year. (iii) Find an expression that relates and , and explain why it represents linear decay.

. Form with negative slope , constant decrease shows linear decay.

3. The initial population of a village is 750. Every year, 50 people move from a nearby city to the village. (i) Find the population of the village after 6 years.

. After 6 years: .

3. The initial population of a village is 750. Every year, 50 people move from a nearby city to the village. (ii) Make a table of values for varying from 0 to 10 years and show how the population, , increases every year.

:0→750,1→800,2→850,3→900,4→950,5→1000,6→1050,7→1100,8→1150,9→1200,10→1250. Increases by 50 each year.

3. The initial population of a village is 750. Every year, 50 people move from a nearby city to the village. (iii) Find an expression that relates and , and explain why it represents linear growth.

. Positive slope 50, constant yearly increase → linear growth.

4. A telecom company charges for a certain recharge scheme. This prepaid balance is reduced by each day after the recharge. (i) Write an equation that models the remaining balance after using the scheme for days. Explain why it represents linear decay.

. Slope , equal daily decrease → linear decay.

4. A telecom company charges for a certain recharge scheme. This prepaid balance is reduced by each day after the recharge. (ii) After how many days will the balance run out?

Set , days.

4. A telecom company charges for a certain recharge scheme. This prepaid balance is reduced by each day after the recharge. (iii) Make a table of values for varying from 1 to 10 days and show how the balance , reduces with time.

:1→585,2→570,3→555,4→540,5→525,6→510,7→495,8→480,9→465,10→450 rupees.

Think and Reflect

Can you guess what the numbers 20 and 150 in the equation represent?

In , is the additional cost per GB (the slope) and is the fixed monthly fee (the y-intercept), the bill when .

Exercise Set 2.5

1. A learning platform charges a fixed monthly fee and an additional cost per digital learning module accessed. A student observes that when she accessed 10 modules, her bill was . When she accessed 14 modules, her bill was . If the monthly bill depends on the number of modules accessed, , according to the relation , find the values of and .

, . Subtract: , . Then . So .

2. A gym charges a fixed monthly fee and an additional cost per hour for using the badminton court. A student using the gym observed that when she used the badminton court for 10 hours, her bill was . When she used it for 15 hours, her bill was . If the monthly bill depends on the hours of the use of the badminton court, , according to the relation , find the values of and .

, . Subtract: , . Then . So .

3. Consider the relationship between temperature measured in degrees Celsius () and degrees Fahrenheit (), which is given by . Find and , given that ice melts at 0 degrees Celsius and 32 degrees Fahrenheit, and water boils at 100 degrees Celsius and 212 degrees Fahrenheit.

, . Subtract: , . Then . So , equivalently .

Think and Reflect

Identify other points on the line by completing the following table. 1 2 5 7 9 12 20 3 15

Line is . For ; ; ; ; ; ; . All satisfy .

Think and Reflect

Differentiate between the graphs of the equations , and .

Both have y-intercept and meet y-axis at . has positive slope 3, rises to the right representing linear growth. has negative slope -3, falls to the right representing linear decay. They are mirror images in the horizontal direction.

Think and Reflect

Does this help you to conclude anything about the linear equation when is fixed but varies?

When is fixed, all lines have the same slope so they are parallel. Changing shifts the line up or down, changing the y-intercept to without changing steepness.

Exercise Set 2.6

1. Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'. (i) , ,

All have and pass through origin. Slopes are 4,2,1. Larger gives steeper line. All represent growth with increasing steepness.

1. Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'. (ii) , ,

All pass through origin with negative slopes -6,-3,-1. More negative means steeper fall. All represent decay, steepness increases with |a|.

1. Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'. (iii) ,

Both through origin, slopes 5 and -5 equal magnitude opposite sign. One rises steeply, other falls steeply with same steepness, symmetric about x-axis.

1. Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'. (iv) , ,

Same slope 3, so three parallel lines. y-intercepts -1,0,1 shift the line vertically by 1 unit each time.

1. Draw the graphs of the following sets of lines. In each case, reflect on the role of 'a' and 'b'. (v) , ,

First two have same slope -2, so parallel with intercepts -3 and 0. Third has slope 2, opposite sign, not parallel to first two, intercept 3.

End-of-Chapter Exercises

1. Write a polynomial of degree 3 in the variable , in which the coefficient of the term is .

Example: . Any cubic polynomial where coefficient of is is acceptable.

2. Find the values of the following polynomials at the indicated values of the variables. (i) if

2. Find the values of the following polynomials at the indicated values of the variables. (ii) if

3. If we multiply a number by and add to the product, we get . Find the number.

Let number be . . So . Thus .

4. A positive number is 5 times another number. If 21 is added to both the numbers, then one of the new numbers becomes twice the other new number. What are the numbers?

Let smaller be , larger . New numbers and . Given . So , , , . Numbers are 7 and 35.

5. If you have and you save every month, find the amount you have after (i) 6 months. Express this as a linear pattern.

Amount . After 6 months: rupees.

5. If you have and you save every month, find the amount you have after (ii) 2 years. Express this as a linear pattern.

2 years = 24 months. rupees. Pattern , linear growth with slope 250.

6. The digits of a two-digit number differ by 3. If the digits are interchanged, and the resulting number is added to the original number, we get 143. Find both the numbers.

Let tens digit , units , with and . So , . Solving with gives → 85; with gives → 58. Both 58 and 85 satisfy conditions (they are interchanges of each other).

7. Draw the graph of the following equations, and identify their slopes and y-intercepts. Also, find the coordinates of the points where these lines cut the y-axis. (i)

Slope , y-intercept . Cuts y-axis at .

7. Draw the graph of the following equations, and identify their slopes and y-intercepts. Also, find the coordinates of the points where these lines cut the y-axis. (ii)

Write . Slope , intercept or . Cuts y-axis at .

7. Draw the graph of the following equations, and identify their slopes and y-intercepts. Also, find the coordinates of the points where these lines cut the y-axis. (iii)

Write . Slope , intercept . Cuts y-axis at .

7. Draw the graph of the following equations, and identify their slopes and y-intercepts. Also, find the coordinates of the points where these lines cut the y-axis. (iv)

Write . Slope , intercept ≈ -3.67. Cuts y-axis at . Lines (ii) and (iv) have same slope 2, so they are parallel.

8. If the temperature of a liquid can be measured in Kelvin units as K and in Fahrenheit units as °F, the relation between the two systems of measurement of temperature is given by the linear equation . (i) Find the temperature of the liquid in Fahrenheit if the temperature of the liquid is 313 K.

°F.

8. If the temperature of a liquid can be measured in Kelvin units as K and in Fahrenheit units as °F, the relation between the two systems of measurement of temperature is given by the linear equation . (ii) If the temperature is 158 °F, then find the temperature in Kelvin.

. So , , K.

9. The work done by a body on the application of a constant force is the product of the constant force and the distance travelled by the body in the direction of the force. Express this in the form of a linear equation in two variables (work and distance ), and draw its graph by taking the constant force as 3 units. What is the work done when the distance travelled is 2 units? Verify it by plotting it on the graph.

Linear equation (slope 3, through origin). For , units. The point lies on the line , verified on graph.

10. The graph of a linear polynomial passes through the points (1, 5) and (3, 11). (i) Find the polynomial .

Let . , . Subtract: , , . So .

10. The graph of a linear polynomial passes through the points (1, 5) and (3, 11). (ii) Find the coordinates where the graph of cuts the axes.

Cuts y-axis at : → . Cuts x-axis at : → → .

10. The graph of a linear polynomial passes through the points (1, 5) and (3, 11). (iii) Draw the graph of and verify your answers.

Plot points and and join. The line indeed passes through and , confirming intercepts.

11. Let and be two linear polynomials such that: (i) . (ii) The polynomial cuts the x-axis at (3, 0). (iii) The sum is equal to for all real . Find the polynomials and .

From (i) . From (iii) , so . From (ii) : , so , . Solve with : , . Hence , .

12. Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage. Stage 1 Stage 2 Stage 3 (i) Draw the next two stages of the pattern. How many matchsticks will be required at these stages?

Stage 1 uses 6, Stage 2 uses 11, Stage 3 uses 16 (adding 5 each time). Stage 4 will use 21, Stage 5 will use 26 matchsticks. Each new hexagon adds 5 matchsticks because one side is shared.

12. Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage. (ii) Complete the following table. Stage Number 1 2 3 4 5 ... n Number of matchsticks

Stage:1→6,2→11,3→16,4→21,5→26,... n→ .

12. Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage. (iii) Find a rule to determine the number of matchsticks required for the nth stage.

Number = . Linear polynomial in with slope 5.

12. Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage. (iv) How many matchsticks will be required for the 15th stage of the pattern?

For , matchsticks.

12. Look at the first three stages of a growing pattern of hexagons made using matchsticks. A new hexagon gets added at every stage which shares a side with the last hexagon of the previous stage. (v) Can 200 matchsticks form a stage in this pattern? Justify your answer.

Set → → , not an integer. So 200 cannot be exactly a stage. Nearest stages are 199 (n=39) and 201 (n=40).

13. Let and be two linear polynomials such that: (i) The graph of passes through the points (2, 3) and (6, 11). (ii) The graph of passes through the point (4, -1). (iii) The graph of is parallel to the graph of . Find the polynomials and . Also, find the coordinates of the point where these lines meet the x-axis.

Slope of : so . Using → , so . Parallel means . Using point (4,-1) for q: → , so . x-intercepts: → → ; → → .

14. What do all linear functions of the form , , have in common?

They can be written as . All have the same x-intercept at , and y-intercept . They all pass through and have positive slope, differing only in steepness and y-intercept which both equal .